CGA clustering based vector quantization approach for human activity recognition using discrete hidden Markov model

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO. 12(85).2014, VOL. 1

95

CGA CLUSTERING BASED VECTOR QUANTIZATION APPROACH FOR HUMAN
ACTIVITY RECOGNITION USING DISCRETE HIDDEN MARKOV MODEL
1

Nguyen Nang Hung Van1, Pham Minh Tuan1, Tachibana Kanta2
Danang University of Science and Technology; ,
2
Kogakuin University;

Abstract - Activity recognition has been taken great consideration
by many scientists all over the world. However, the conventional
research results need to be improved because of the complexity
and unstability of object recognition. Especially with human activity
recognition (HAR) in 3-dimensional space, the vector quantization
based on k-means was not able to cluster two objects rotating
around a common point but on a different plane because they have
the same cluster centroid. In this paper, we propose a new method
of vector quantization (VQ) performance optimally distribute VQ
codebook components on Hidden Markov Model (HMM) state. This
proposed method is carried out through two steps. First, the
proposed method use Conformal Geometric Algebra (CGA)
clustering algorithms to optimize VQ. Then, the proposed method
uses discrete HMM to recognize the human activity. The
experimental result with the CMU graphics lab motion capture
database shows that the proposed method is more effective than
conventional method.
Key words - Hidden Markov Model; vector quantization; clustering;

k-mean; conformal geometric algebra.

1. Introduction
Human activity recognition is one of the important
areas of computer vision research. Its applications include
intelligent security monitoring system, health care systems,
intelligent transportation systems, and a variety of systems
that involve interactions between people and electronic
devices such as human computer interfaces. Today there
are many researches on human activity recognition area.
For example, the discrete HMM (DHMM) is one of the
most common recognition models and it is applied in many
human activity recognitions such as human activity
recognition using monocular camera [1] or speech
recognition system [2].
In this paper, we focus on clustering algorithm based VQ
for DHMM [3]. In conventional methods, the k-means is
usually used to quantize a vector before applying to DHMM.
The advantage of k-means algorithm is simple and easy
to understand and install. It is able to apply to assign the data
to groups using Euclidean distance. However, using
Euclidean distance is also the disadvantage of k-means
algorithm in the case of 3-dimensional data. For example,
when we have two objects rotating around a common point
but is not same a plane, we can not cluster the coordinates of
two objects correctly because two cluster centers of k-means
will be same. Therefore, the result of k-means based vector
quantization for the 3-dimensional rotation data such as
human activity is not good. So, this paper proposed to use
CGA clustering to quantize a vector for DHMM. CGA is a

part of (Geometric Algebra) GA and is also called Clifford
Algebra. CGA is the GA constructed over the resultant space
of a projective map from an m-dimensional Euclidean or
pseudo-Euclidean base space ℛ𝑚 into 𝒢𝑚+1,1 . This allows
operations on the m-dimensional space, including rotations,

translations and reflections to be represented using versors
of the GA [4]; and it is found that points, lines, planes, circles
and spheres gain particularly natural and computationally
amenable representations [5, 6]. And, there are many
applications of GA as signal processing model, using image
processing of complex spatial GA [7] or quaternions [8].
In this paper, we present a new CGA clustering
approach to improve the accuracy of the DHMM on HAR
system based on VQ by implementing the optimal
distribution of the codebook of HMM states. This
technique, which has been named the distributed VQ of
HMM is done through two steps. The first is to use CGA
clustering algorithm to optimize the VQ, the next step will
be to conduct HMM parameter estimation and
classification of action [9].
The paper is structured as follows. The first is the
introduction of this paper. The second presents the related
research. Section 3 reports conformal geometric algebra and
describes CGA clustering approach for DHMMs. Section 4
reports the comparative results of the proposed method using
CGA clustering and conventional methods using k-means.
Finally, the 5th section summarizes this paper.
2. Related research
This section presents the basic of a VQ for the discrete

hidden Markov models (DHMMs). This section
summarises a k-means based VQ and review DHMMs.
2.1. K-means based vector quantization
Vector quantization is a process of the mapping of a
sequence of 𝑚-dimensional continuous vectors [10, 11]
𝑶 = {𝒗1 , ⋯ , 𝒗 𝑇 }, 𝒗𝑡 ∈ 𝐑𝑚 to a discrete, one dimensional
̂ = {𝒗
̂1 , ⋯ , 𝒗
̂ 𝑇 }, 𝒗
̂𝑡 ∈ 𝐍
sequence of codebook indices 𝑶
where a codebook 𝑪 = {𝒄1 , ⋯ , 𝒄𝐾 }, 𝒄𝑘 ∈ 𝐑𝑚 and 𝐾 is the
number of centroids 𝒄𝑖 . The assignment of the continuous
sequence to the codebook indices is a minimum distance
search if the codebook 𝑪 is generated,
̂𝑖 = argmin 𝑑(𝒗𝑖 , 𝒄𝑘 ), ∀𝑖 ∈ [1, ⋯ , 𝑇]
𝒗
𝑘

where 𝑑(𝒗𝑖 , 𝒄𝑘 ) = ‖𝒗𝑖 − 𝐜𝑘 ‖2 is the squared Euclidean
distance. There are many ways to generate the codebook.
This section describes a basic method to generate 𝑪 using
k-means clustering.
Given a training set 𝑺𝒕𝒓𝒂𝒊𝒏 = {𝑶1 , ⋯ , 𝑶𝑁 }, where
𝑵 = |𝑺𝒕𝒓𝒂𝒊𝒏 |
is
the
number
of
samples

𝑶𝑖 = {𝒗𝑖,1 , ⋯ , 𝒗𝑖,𝑇 }, 𝒗𝑖,𝑡 ∈ 𝐑𝑚 . A codebook 𝑪 is calculated
by minimizing the following problem,
𝐾

𝑁

𝑇

min ∑ ∑ ∑ 𝑢𝑘,𝑖,𝑡 𝑑(𝒗𝑖,𝑡 , 𝒄𝑘 )
𝑢,𝐜

𝑘=1 𝑖=1 𝑡=1


96

Nguyen Nang Hung Van, Pham Minh Tuan, Tachibana Kanta
𝐾

s. t.

∑ 𝑢𝑘,𝑖,𝑡 = 1 ,

𝑢 𝑘,𝑖,𝑡 ∈ {0, 1},

𝑘=1
2

where 𝑑(𝒗𝑖,𝑡 , 𝒄𝑘 ) = ‖𝒗𝑖,𝑡 − 𝐜𝑘 ‖ is the squared
Euclidean distance between the vector 𝒗𝑖,𝑡 and the kth

codebook centroids 𝒄𝑘 . The k-means clustering algorithm
to calculate the codebook 𝑪 is described as follows.
algorithm kmeans_codebook()
input:
v[N][T_N]: training set
K: number of centroids
output:
u[N][T_N]: memberships
C[K]: array of codebook centroids
begin
δ  1
while (δ>0)
δ  0
for k from 0 to K-1 do
C_new[k]  0 //Zero vector
C_size[k]  0
endfor
for i from 0 to N-1 do
for t from 0 to T(i)-1 do
dmin  ∞
n  0
for k from 0 to K-1 do
d  |V[i][t] – C[k]|
if ddmin  d
n  k
endif
endfor
if u[i][t] ≠ n then
δ  δ + 1

u[i][t]  n
endif
C_new[n]  C_new[n] + V[i][t]
C_size[n]  C_size[n] + 1
endfor
endfor

(3) to calculate the probability that a given sequence of
outputs originated from the system.
This paper focuses on ability (3) of discrete HMMs. It
means that this paper uses DHMMs for sequence
classification. The DHMMs have been defined by the
following set of parameters,
𝜆 = {𝐴, 𝐵, 𝜋}
where 𝐴 is the state transition probability distribution
given in the form of a matrix 𝐴 = {𝑎𝑖𝑗 }. 𝐵 is the
observation symbol (codebook index) probability
distribution given in the form of a matrix 𝐵 = {𝑏𝑗 (𝑘)} and
𝜋 is the initial state distribution. Figure. 1 shows an
example of Hidden Markov model.
a22
𝜋2
a12
a21

a11

𝜋3

a31

b11 b12 b13

b31 b32 b33

b21

b22

v1

v32

b23
v3

Figure 1. An example of Hidden Markov model

Training data
class 1

class 2

1
….
1

1
….
0

1
….
0

0
….
0

λ

λ

2.2. Discrete HMMs

(1) to infer the most likely sequence of states that
produced a given output sequence,
(2) to infer which will be the most likely next state,

a33

a13

𝜋1

for k from 0 to K-1 do
C[k]  C_new[k] / C_size[k]
endfor
endwhile
end

HMMs are the important methods to model temporal
and sequence data. They are especially known for their
application in real time pattern recognition such as
handwriting digits recognition, speech recognition [12] and
human ativity recognition. HMMs attempt to model such
systems and allow:

a23

a32

recognition
:

O = {v1, … ,vT}

P(O/λ)

P(O/λ)

Figure 2. Recognition via HMM

In order to use DHMMs, the continuous observations
𝑶 = {𝒗1 , ⋯ , 𝒗 𝑇 }, 𝒗𝑡 ∈ 𝐑𝑚 are vector quantized yielding
̂ = {𝒗
̂1 , ⋯ , 𝒗
̂ 𝑇 }, 𝒗
̂𝑡 ∈ 𝐍
discrete observation sequences 𝑶

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO. 12(85).2014, VOL. 1

using the codebook. In the case of classification problem,
we need create a set of models and specialise each model
to recognize each of the separated classes. The parameters
𝜆𝑖 of the ith model can be trained with the well known EM
algorithm [13]. After all models have been trained, the
probability of the unknown-class sequence can be
computed for each model. As each model specialised in a
given class, the one which outputs the highest probability
can be used to determine the most likely class for the new
sequence, as shown in Figure 2.
3. Proposed method
This paper proposed a new vector quantization
approach for DHMMs based on conformal geometric
algebra clustering. This section reviews conformal
geometric algebra and describes conformal geometric
algebra clustering approach for DHMMs.
3.1. Conformal Geometric Algebra
Conformal Geometric Algebra is a part of Geometric
Algebra [14] and is also called Clifford Algebra. GA
defines the signature 𝑝 + 𝑞 orthonormal basis vector 𝒪 =
{𝑒1 , … , 𝑒𝑝 , 𝑒𝑝+1 , … , 𝑒𝑝+𝑞 }, such as 𝑒𝑖2 = 1, ∀𝑖 ∈ {1, … , 𝑝}
and 𝑒𝑖2 = −1, ∀𝑖 ∈ {𝑝 + 1, … , 𝑞}. GA denotes 𝒪 by 𝒢𝑝,𝑞 .
For example, m-dimensional Euclidean vector space ℛ𝑚 is
denoted by 𝒢𝑚,0 .
A CGA space is extended from the real Euclidean
vector space ℛ 𝑚 by adding 2 orthonormal basis vector.

Thus, a CGA space is defined by 𝑚 + 2 basis vectors 𝒪 =
{𝑒1 , … , 𝑒𝑚 , 𝑒+ , 𝑒− }, where 𝑒+ and 𝑒− are defined as
following:
𝑒+2 = 𝑒+ ∙ 𝑒+ = 1,
𝑒−2 = 𝑒− ∙ 𝑒− = −1,
𝑒+ ∙ 𝑒− = 𝑒+ ∙ 𝑒𝑖 = 𝑒− ∙ 𝑒𝑖 = 0, ∀𝑖 ∈ {1, … , 𝑚}.
Thus, a CGA can be expressed by 𝒢𝑚+1,1 . In addition,
CGA defined:
1
𝑒0 = (𝑒− − 𝑒+ )
2
𝑒∞ = (𝑒− + 𝑒+ ).
It is easy to see that:
𝑒0 . 𝑒0 = 𝑒∞ . 𝑒∞ = 0,
𝑒0 . 𝑒∞ = 𝑒∞ . 𝑒0 = 0,
𝑒0 . 𝑒𝑖 = 𝑒∞ . 𝑒𝑖 = 0, ∀𝑖 ∈ {1, … , 𝑚}.
A conformal vector 𝑆 is generally written in the
following:
𝑆 = 𝒔 + 𝑠∞ 𝑒∞ + 𝑠0 𝑒0
Where 𝒔 = ∑𝑚
𝑖 𝑠𝑖 𝑒𝑖 is a real vector in the Euclidean
space ℛ𝑚 . And 𝑠∞ , 𝑠0 are the scalar coefficients of the basic
vector 𝑒∞ and 𝑒0 .The CGA can express a point, a sphere or
a plane based on 𝑆. For example, sphere is represented as a
following conformal vector:
1
𝑆 = 𝑥 + {‖𝒙‖2 − 𝑟 2 }𝑒∞ + 𝑒0 ,
2
where the sphere has center 𝒙 and radius 𝑟 in real
Euclidean space ℛ𝑚 . Note that the inner product 𝑆 ∙ 𝑄 is 0

for any point 𝑄 on the surface of the sphere 𝑆.

97

3.2. CGA clustering based vector quantization
This paper proposes a new vector quantization
approach for DHMMs based on conformal geometric
algebra clustering.
Given a training set 𝑺𝒕𝒓𝒂𝒊𝒏 = {𝑶1 , ⋯ , 𝑶𝑁 }, where 𝑵 =
|𝑺𝒕𝒓𝒂𝒊𝒏 |
is
the
number
of
samples
𝑶𝑖 =
{𝒗𝑖,1 , ⋯ , 𝒗𝑖,𝑇 }, 𝒗𝑖,𝑡 ∈ 𝐑𝑚 . This paper converts all samples
𝑶𝑖 to a set of points 𝑷𝑖 = {𝒑𝑖,1 , ⋯ , 𝒑𝑖,𝑇 }, 𝒑𝑖,𝑡 = 𝒗𝑖,1 +
1

2

‖𝒗𝑖,1 ‖ 𝑒∞ + 𝑒0 ∈ 𝒢𝑚+1,1 in CGA space. The codebook is
defined by a set of vector 𝑪 = {𝒄1 , ⋯ , 𝒄𝐾 }, 𝒄𝑘 = 𝒔𝑘 +
𝑠𝑘,∞ 𝑒∞ + 𝑠𝑘,0 𝑒0 ∈ 𝒢𝑚+1,1 and is calculated by minimizing
the following problem,
2

𝐾

𝑁

𝑇

min ∑ ∑ ∑ 𝑢𝑘,𝑖,𝑡 𝑑(𝒑𝑖,𝑡 , 𝒄𝑘 )
𝑢,𝐜

𝑘=1 𝑖=1 𝑡=1
𝐾

s. t.

∑ 𝑢𝑘,𝑖,𝑡 = 1 ,

𝑢 𝑘,𝑖,𝑡 ∈ {0, 1},

𝑘=1
1

2

2

where 𝑑(𝒑𝑖,𝑡 , 𝒄𝑘 ) = (𝒗𝑖,𝑡 ∙ 𝐬𝑘 − 𝑠𝑘,∞ − ‖𝒗𝑖,1 ‖ 𝑠𝑘,0 )
2
is the squared distance between the point 𝒑𝑖,𝑡 and the kth
codebook centroids 𝒄𝑘 in CGA space.
CGA based clustering proceeds by alternating between
two steps:
• Assignment step: Assign each observation to the cluster

whose mean yields the least within-cluster sum of
squared distance in CGA space;
• Update step: Calculate the new means to be the
centroids of the observations in the new clusters.
The centroid 𝒄𝑘 can be calculated by minimization the
following L fuction:
𝑁

𝑇

2
1
2
𝐿 = ∑ ∑ 𝑢𝑘,𝑖,𝑡 ((𝒗𝑖,𝑡 ∙ 𝐬𝑘 − 𝑠𝑘,∞ − ‖𝒗𝑖,1 ‖ 𝑠𝑘,0 )
2
𝑖=1 𝑡=1

− 𝜆(‖𝐬𝑘 ‖2 − 1))
The CGA clustering algorithm to calculate the
codebook 𝑪 is described as following.
algorithm cgaclustering_codebook()
input:
v[N][T_N]: training set
K: number of centroids
output:
u[N][T_N]: memberships
C[K]: array of codebook centroids in CGA
means
Begin
δ  1

while (δ>0)
δ  0
for i from 0 to N-1 do
for t from 0 to T(i)-1 do
dmin  ∞
n  0
for k from 0 to K-1 do


98

Nguyen Nang Hung Van, Pham Minh Tuan, Tachibana Kanta

d  |V[i][t] – C[k]|
if ddmin  d
n  k
endif
endfor
if u[i][t] ≠ n then
δ  δ + 1
u[i][t]  n
endif
endfor
endfor

In this experiment, we defined a sequence of 𝑚dimensional continuous vectors 𝑶 by using the direction of
6 segments. The name list of 6 segments shows as Figure. 3.
4.2. Experiment result
In this section, we demonstrate the performance of our

proposed method by using running data and walking data
downloaded from the CMU graphics lab motion capture
database website. We compare our proposed method with
k-means based VQ. Figure. 3 shows the training model and
recognition using CGA Clustering based VQ approach for
DHMM human activity recognition.

for k from 0 to K-1 do
C[k]  argminL(k)
endfor
endwhile
end

4. Experiment Result
The results are achieved by conducting feature
selection, discretization data and experiments of HAR
using the CMU graphics lab motion capture database
4.1. CMU graphics lab motion capture database
CMU graphics lab motion capture database [15]
includes the data set made by a Vicon motion capture
system consisting of 12 infrared MX-40 cameras, each of
which is capable of recording at 120 Hz with images of 4
megapixel resolution. Motions are captured in a working
volume of approximately 3m x 8m. The capture subject
wears 41 markers and a stylish black garment. Vicon
software will create two data files: ASF file and AMC file.

Figure 4. Discrete data model

• In the ASF file (Acclaim Skeleton File), a base pose

is defined for the skeleton that is the starting point
for the motion data. ASF has information: length,
direction, local coordinate frame, number of Dofs,
joint limits and hierarchy, connections of the bone.
• The AMC file (Acclaim Motion Capture) contains
the motion data for a skeleton defined by an ASF
file. The motion data is given a sample at a time.
Each sample consists of a number of lines, a
segment per line, containing the data.
Figure 4. Discrete data model

For training and testing the model, we have a data set
with 46 trials of the human running action and 131 trials of
the human walking action. We separated the data in two
equal parts by randomly. The first part is used for training
and the second is used for testing.
Table 1. Comparison results between K-means based VQ and
CGA clustering based VQ.

Figure 3. Name list of 6 segments.

Class
num
2
3
4
5
2

Frames per K-means (%) CGA clustering (%)

second
24
55 (%)
86 (%)
24
71 (%)
91 (%)
24
53 (%)
94 (%)
24
49 (%)
94 (%)
12
47 (%)
26 (%)


ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO. 12(85).2014, VOL. 1

3
4
5

12
12
12

60 (%)
54 (%)

68 (%)

82 (%)
77 (%)
63 (%)

The Table. 1 shows that the accuracy of human activity
recognition. The result shows that the proposed method
using CGA clustering based VQ was better than k-means
based VQ. The accuracy of recognition using proposed
method was the highest (94%) in the case of class number
is 4 and using the data with 24 frames per second.
5. Conclusions
This paper presented the basic of a VQ for DHMMs.
This paper also summarised a k-means based VQ and
reviewed DHMMs. Then, this paper proposed a new
approach of CGA Clustering based VQ for HMM. The
experiment result of human activity recognition using
CMU graphics lab motion capture database showed that the
proposed method is better than conventional k-means
based VQ method.
From this result, we can use CGA clustering algorithm
to instead of k-means algorithm in the field of speech
recognition, action recognition of objects. At the same
time, the results of research opens up a new research
direction in recognition theory, automatic control systems
[16] and a variety of systems that involve interactions
between people and electronic devices such as human
computer interfaces.
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(The Board of Editors received the paper on 26/10/2014, its review was completed on 30/10/2014)